LOS ANGELES — A serial-killer mystery in which the culprit turns out to be one of Satan’s minions, Scott Derrickson’s “Deliver Us From Evil” adapts the detective genre to an exorcism tale that is very serious about the prospect of demonic possession. Though based on claims made by real-life NYPD officer-turned-paranormal investigator Ralph Sarchie (played here by Eric Bana), the picture is stolen by a fictional character — a composite religious figure played with a predictable level of smolder by “Carlos” star Edgar Ramirez.

More aesthetically coherent if less frightening than the director’s 2012 hit “Sinister,” the pic has a shot at surpassing that outing commercially by virtue of its appeal to devout Catholics. That audience will find “Evil” very respectful of their faith, though its nods to religion are genre-appropriate and never preachy enough to alienate the average horror fan.

Sarchie, a cop who covers the South Bronx with partner Butler (Joel McHale), is proud of an internal “radar” that steers him away from mundane radio calls and toward the juicy cases. But an incident in which a deranged mother threw her toddler into the Bronx Zoo’s lion pit is juicier than usual, with a mysterious, blood-stained bystander disrupting the investigation before vanishing. Still, Sarchie is ready to write the mother off as a garden-variety psycho even when a Jesuit priest, Ramirez’s Mendoza, shows up to insist on a more complicated explanation.

Only after other crimes prove to share both tangible connections (tracing to three Marines who had a disturbing encounter during the Iraq war) and supernatural overtones does Sarchie admit there may be something to this chain-smoking, whiskey-drinking priest’s claim. Maybe, the lapsed-Catholic cop starts to worry, the man they’re hunting really is possessed.

Enjoyably, this particular case isn’t limited to one spirit-controlled victim terrorizing his family and friends. “Poltergeist-y” aftereffects haunt the places and people he has visited — even Sarchie, who is plagued by shock-cut hallucinations and whose house has become a very scary place for his young daughter. (Olivia Munn, playing Sarchie’s wife, tries to comfort the poor girl, but the film ratchets up her bump-in-the-night torment.)

The investigation yields some appreciably icky encounters with putrefying corpses and deranged prisoners, which is good considering the sometimes flimsy cop-movie stuff that surrounds the scares. Bulked-up McHale, who acquits himself well in action scenes, gets the kind of jadedly quippy dialogue one can easily imagine him parodying on “Community”; Munn’s part could have been scripted by cut-and-pasting any of a hundred other neglected-policeman’s-wife characters.

While the supernatural side of the film suffers a flaw or two — continued references to The Doors are superfluous and sometimes chuckle-inducing — its central conflict works. Ramirez, shaggy enough to be the Serpico of exorcists but exuding calm wisdom instead of obsessive determination, makes faith look cool. His seriousness enables the movie’s desire to dig into the mechanics of the climactic exorcism — though Sean Harris, drooling and growling and threatening as the possessed man, deserves credit as well. Derrickson’s FX crew careens along the dividing line between excitement and silly bombast, conjuring hurricane-like forces and gory transformations while the holy man attempts to rescue an innocent mortal from his otherworldly tormentors.

“Deliver Us From Evil,” a Sony release, is rated R by the Motion Picture Association of America for “bloody violence, grisly images, terror throughout, and language.” Running time: 118 minutes.